The present invention relates to a wavelength conversion device in which a laser beam is applied on a nonlinear optical medium so as to obtain a beam of a different wavelength.
In the prior art, the laser generates a coherent beam of sharp directivity and is widely applied in various fields such as material processing and measurement fields. Recently, it has also been applied in medical and chemical industry fields. However, with the exception of some types, the laser vibrates or oscillates only at a specific wavelength which results in one of the most serious obstacles to its applications.
In order to solve this problem, wavelength conversion technology using various kinds of nonlinear optical materials is being used. Generally, when a beam is applied on a transparent crystal, dipoles are excited by polarization due to a vibrating electric field and the vibrating dipoles generate a new beam. The relationship between the polarization P and the electric field E is expressed by: EQU P=P.sub.L +P.sub.NL EQU P.sub.L =.chi..sup.(1) .multidot.E EQU P.sub.NL =.chi..sup.(2) :E.multidot.E+.chi..sup.(3): E.multidot.E.multidot.E+.multidot..multidot..multidot. (1)
wherein P.sub.L is a linear polarization, P.sub.NL is a nonlinear polarization and .chi..sup.(1), .chi..sup.(2), .chi..sup.(3) are primary, secondary and tertiary polarizations, respectively. Generally, a high order polarization is smaller than .chi..sup.(1) and therefore, the higher order terms in the equation (1) may be disregarded in the case of a normal beam. However, when the laser beam is powerful and has a large E-value, a higher order polarization is present. Especially, as the secondary term is larger than the tertiary term and downward, we will express P.sub.NL in the following equation by secondary terms only. Accordingly, when EQU P.sub.NL =(P.sub..chi., P.sub.y, P.sub.z).sup.T, EQU E=(E.sub..chi., E.sub.y, E.sub.z).sup..tau., ##EQU1## will follow and in this equation, the x, y and z axes are taken in the direction of axis of the crystal and the z-axis is taken in the direction of the optical axis.
Now, assuming that there are two electric fields E.sub.1 and E.sub.2 having angular frequencies .omega..sub.1, .omega..sub.2 present in a crystal, and in that case, if EQU E=E.sub.1 +E.sub.2 EQU E.sub.1 =E.sub.10 cos (.omega..sub.1 t-k.sub.1 .multidot.r+0.sub.1)(3) EQU E.sub.2 =E.sub.20 cos (.omega..sub.2 t-k.sub.2 .multidot.r+0.sub.2)
then the following equation will follow: ##EQU2## wherein r is a position vector, k.sub.1, k.sub.2 are wave number vectors and 0.sub.1, 0.sub.2 are phase angles. These terms can be rewritten as follows: EQU cos.sup.2 (.omega..sub.i t-k.sub.i .multidot.r+0.sub.i) =1/2{1+cos (2.omega..sub.i t-2k.sub.i .multidot.r+20.sub.i)}(i=1,2) (5) ##EQU3## As a result, P.sub.NL may be divided into frequency components EQU 0, 2.omega..sub.1, .omega..sub.1 -.omega..sub.2, .omega..sub.1 +.omega..sub.2, 2 .omega..sub.2.
Of these components, the zero frequency component, that is, a DC component, means the rectification of an optical frequency electromagnetic wave in the nonlinear medium. Further, the 2.omega..sub.2 and 2.omega..sub.2 components are polarizations for causing second harmonic generation, .omega..sub.1 -.omega..sub.2 are those for causing differential frequency generation, .omega..sub.1 +.omega..sub.2 are those for causing sum frequency generation, respectively. However, it does not always follow that beams generate from all of these polarization waves of different frequencies; the kinds of frequency beams generated from the polarization waves depend on the phase matching conditions to be described below. Further, not only the incident beam, but also the interaction between it and the beam actually generated must be taken into consideration.
Assuming that a third beam of angular frequency .omega..sub.3 generates from the above two beams and the electric field thereof is E.sub.3, the entire electric field will be: EQU E=E.sub.1 +E.sub.2 +E.sub.3 ( 7)
and accordingly, the secondary nonlinear polarization will include a total of 9 frequency components as expressed by the following equation: ##EQU4## Where the generated beam is of a sum frequency .omega..sub.3 =.omega..sub.1 +.omega..sub.2, EQU P.sup.(.omega..sbsp.3.sup.-.omega..sbsp.2.sup.) =P.sup.(.omega..sbsp.1.sup.) =P.sup.(.omega..sbsp.1), P.sup.(.omega..sbsp.3.sup.-.omega..sbsp.1 .sup.) =P.sup.(.omega..sbsp.2.sup.) P.sup.(.omega..sbsp.1.sup.+.omega..sbsp.2.sup.) =P.sup.(.omega..sbsp.3.sup.)
will follow so that the term E.sub.1 is derived from the terms E.sub.3 and E.sub.2, the term E.sub.2 from the terms E.sub.3 and E.sub.1 and the term E.sub.3 from the terms E.sub.1 and E.sub.2. That is, the three modes of .omega..sub.1, .omega..sub.2, .omega..sub.3 are combined.
the relationship between the polarization P.sub.NL and the electrostatic field E is expressed by the Maxwell equation. That is ##EQU5## as further, c is the velocity of beam in a vacuum, .eta. is the refractive index of the medium and .mu. is the magnetic permeability of the medium. In this case, the field vectors of the three beams are expressed by the following equation on the assumption that they progress in the z-direction. EQU E.sub.i =e.sub.i .rho..sub.i cos(.omega..sub.i t-k.sub.i z+0.sub.i) (i=1,2,3) (10)
wherein e.sub.i is a unit vector and .rho..sub.i is a variable representing an amplitude. When the equations (7), (8) and (10) are substituted into the equation (9) and when it is assumed that EQU d.sup.2 .rho..sub.i /d.sub.z.sup.2 &lt;&lt;k.sub.i .multidot.d.rho.i/dz,
then the following equation will result. ##EQU6## wherein EQU K-.chi..sup.(2) .mu..sub.0 /2, .DELTA.K=k.sub.3 -k.sub.1 -k.sub.2, EQU .theta.=.DELTA.kz-0.sub.3 +0.sub.2 +0.sub.1
provided that .chi..sup.(2) which is an element of a matrix .chi..sup.(2) differs depending on how the plane of polarization for the incident beam is oriented. Further, .mu..sub.0 is the magnetic permeability in a vacuum. By solving this differential equation, it is possible to obtain variations of the amplitudes .rho..sub.1, .rho..sub.2 and .rho..sub.3 of the three frequency components.
Now let us consider the Second Harmonics Generation (SHG) as a particular case. This case corresponds to the equation (11) provided that EQU .omega..sub.1 =.omega..sub.2 =.omega., .omega..sub.3 =.omega..sub.1 +.omega..sub.2 =2.omega., and .rho..omega.=.rho..sub.1 =.rho..sub.z, .rho..sub.2.omega. =.rho..sub.3
and the following equation is derived therefrom. ##EQU7## To obtain the variation of .rho..sub.2.omega. by solving the above equation, the result shown in FIG. 7 will be obtained. In this case, EQU .DELTA.S=.DELTA.kl.sub.o
and the l.sub.0 designates the length required for about 75% of the fundamental wave output to be converted to a double high harmonic wave when .DELTA.k=0. From this result, it will be understood that when .DELTA.k=0, .rho..sub.2.omega. increases uniformly together with z but when .DELTA.k .noteq.0, .rho..sub.2.omega. vibrates and the wavelength conversion is not effectively performed. From a qualitative point of view, this means that when .DELTA.k=0, the second high harmonic wave is effectively amplified because the polarization wave and second high harmonic wave generating therefrom go side by side in the same phase but when .DELTA.k .noteq.0, a phase mismatch takes place because the velocity of polarization wave differs from that of the second high harmonic wave.
Now, the conditions for satisfying .DELTA.k=0 will be described below.
k.sub..omega. and k.sub.2.omega. are expressed by: ##EQU8## wherein .lambda..sub..omega., .lambda..sub.2.omega. and .eta..sub..omega., .eta..sub.2.omega. are the lengths and refractive indices of the fundamental wave and second high harmonic waves, respectively. Therefore, to satisfy .DELTA.k=0, .eta..sub..omega. =.eta..sub.2.omega. must be satisfied. However, as will be understood from the fact that a white beam is separated by a prism, the refractive index differs depending on the wavelength and generally, the relationship of .eta..sub.2.omega. &gt;.eta..sub..omega. is established. That is, .DELTA.k.noteq.0 is usual.
To solve this problem, various kinds of phase matching means have been proposed. The first of them is the use of a nonlinear optical crystal having an optical anisotropy. In such crystal, the incident beam is divided into normal and abnormal beams and the refractive index of the former is constant irrespective of the direction of incidence while that of the latter changes depending on the direction of incidence. However, there is a direction in which the refractive indices of both of the beams coincide with each other and this direction is called an optical axis. Further, where the crystal has a single optical axis, it is called a uniaxial crystal and where two, it is called a biaxial crystal. Here, we consider only the uniaxial crystal. Note that the two beams displace from each other except in the direction of the optical axis and a direction normal thereto. For example, where the angle of incidence is 0.degree. (the angle between the surface of the crystal and the incident beam is 90.degree.), the normal beam progresses linearly but the abnormal beam refracts as shown in FIG. 8. In summation, it is in the optical axis direction that the refractive indices of the normal and abnormal beams are equal and in the direction normal to the optical axis that the progressing directions of the normal and abnormal beams are equal.
FIG. 9 shows a relationship between the directions of a beam and .eta. wherein .eta..sup.o.sub..omega. and .eta..sup.e.sub..omega. are the refractive indices of fundamental wave normal and abnormal beams, and .eta..sup.o.sub.2.omega. and .eta..sup.e.sub.2.omega. are those of second high harmonic wave normal and abnormal beams, respectively. From this figure, it will be understood that at an angle of .theta..sub.m with respect to the optical axis (z axis) .eta..sup.o.sub..omega. (.theta..sub.m) and .eta..sup.e.sub.2.omega. (.theta..sub.m) coincide with each other. That is, phase matching is satisfied between fundamental wave as normal beam and second high harmonic wave generated as abnormal beam when the fundamental wave as normal beam is applied at angle .theta..sub.m. To make the fundamental wave a normal beam, it may be applied as a linear polarization in a direction normal to the z axis whereby a high harmonic wave is emitted as a linear polarization parallel to the z axis. A case in which the fundamental wave is a normal beam and the high harmonic wave is an abnormal beam is generally called a Type I phase matching. On the other hand, it is possible for the fundamental wave to be a combination of a normal beam and an abnormal beam and such case is called a Type II phase matching. For the sake of simplicity of explanation, we will consider only the Type I phase matching.
The above phase matching (i.e., the Type I) is convenient because it is performed by mere adjustment of the incidence angle. However, as shown in FIG. 8, since the fundamental wave and the second high harmonic wave progress in different directions, the region of interaction of both the waves is limited to the hatched portion in the figure. However, if .theta..sub.m could be 90.degree., the progressing directions of both normal and abnormal beams will coincide so that a high efficiency wavelength conversion can be made but it is general that .eta..sup.o.sub..omega..noteq..eta..sup.e.sub.2.omega. (90.degree.). However, there is a case in which both of the directions coincide when the temperature of the crystal is changed. This is because the temperature dependability differs between the refractive indices .theta..sup.o.sub..omega. and .eta..sup.e.sub.2.omega.. For example, when the output of the YAG laser (of wavelength of 1.06 .mu.m) is a fundamental wave, it is possible to obtain a phase matching of 90.degree. by controlling the temperature to 165.degree. C. for lithium niobate (LiNb0.sub.3) and 181.degree. C. for potassium niobate (KNbO.sub.3). Of course such temperatures differ depending on the wavelength of the laser output so that for example, it is possible to obtain the 90.degree. phase matching at 25.degree. C. with respect to a wavelength of 0.86 .mu.m by using KNbO.sub.3.
As other approaches for performing phase matching, there is one that uses a wave guide and one that applies fundamental waves from different directions. The former is represented by an optical fiber and a thin film wave guide and makes use of the phenomenon that the effective refractive index changes when the diameter or thickness of the wave guide is changed. That is, the mode of the beam propagating through the wave guide is dispersive and the dependability of the refractive indices of the beam with respect to the width of the wave guide differs. Therefore, phase matching is performed by allotting the fundamental and high harmonic waves to different modes and by adjusting the thickness of the wave guide. On the other hand, the latter approach takes into account phase matching as a vector volume. Heretofore, it has been considered that the fundamental and high harmonic waves progress in the same direction. However, as shown in FIG. 10, when two fundamental waves are applied from different directions with wave number vectors K.sub.107 .sup.1 and k.sub..omega..sup.2, the induced nonlinear polarization wave propagates with a wave number vector of k.sub..omega..sup.1 +k.sub..omega..sup.2. The condition for phase matching between the nonlinear polarization wave and the resultant second high harmonic wave of a wave number vector k.sub.2.omega. which propagates in the same direction as the former is expressed by the following equation when .vertline.k.sub..omega..sup.1 .vertline.=.uparw.k.sub..omega..sup.2 .vertline.: EQU .eta..sub..omega. cos .alpha.=.eta..sub.2.omega. ( 15)
Accordingly, in the case of a bulk crystal, phase matching is performed by adjusting o when the fundamental waves are normal beams and the second high harmonic wave is an abnormal beam. Further, it is also possible to perform a phase matching by using a wave guide such that the fundamental and high harmonic waves are allotted to different modes and .alpha. is adjusted instead of the width of the wave guide.
Of the above-mentioned conventional phase matching approaches, the one in which the incidence angle to the nonlinear optical crystal is adjusted is convenient but it has been accompanied with the problem that the interaction of the fundamental wave and high harmonic wave cannot be maintained long due to double refraction. On the other hand, the approach in which the temperature of the crystal is adjusted by applying the beam in the direction normal to the optical axis makes it possible to perform a high efficiency wavelength conversion, since the directions of the fundamental wave and high harmonic wave coincide with each other, but since the range in which the refractive index varies by changing the crystal temperature is quite small, the refractive index value must originally be suitable for the range so that the means is applicable to only a few kinds of crystals. Further, the wavelength range for effecting 90.degree. phase matching is naturally limited.
Likewise, the approach of adjusting the width of the wave guide is hopeful but it still has many difficulties in actual practice because the wave guide width must be controlled to a high degree of accuracy.
Further, the approach of applying two fundamental waves from two different directions has had the problem that when the fundamental waves are so converged as to increase their energy density, the region where they overlap becomes small resulting in decreasing the efficiency.
An object of the present invention is to eliminate the above-mentioned problems and to provide a high efficiency wavelength conversion device which is applicable to a variety of nonlinear optical materials and to beams in a wide range of wavelengths.